Hence, we see that if the system is stable, then H(z) converges for z = ejω. That is, for a stable discrete-time LTI system, the ROC of H(z) must contain the unit circle |z| = 1.
13. The ROC of the z-transform of any signal cannot contain poles.
A. True
B. False
Answer: A
The ROC of the z-transform of any signal cannot contain poles. Since the value of the z-transform tends to infinity, the ROC of the z-transform does not contain poles.
14. Is the discrete-time LTI system with impulse response h(n) = an(n) (|a| < 1) BIBO stable?
A. True
B. False
Answer: A
Given h(n) = an(n) (|a|<1)
The z-transform of h(n) is H(z) = \(\frac{z}{z-a}\), ROC is |z|>|a|
If |a|<1, then the ROC contains the unit circle. So, the system is BIBO stable.
15. What is the ROC of a causal infinite length sequence?
A. |z|<r1
B. |z|>r1
C. r2<|z|<r1
D. None of the mentioned
Answer: B
The ROC of the causal infinite sequence is of form |z|>r1 where r1 is the largest magnitude of poles.
16. Which of the following justifies the linearity property of z-transform?[x(n)↔X(z)].
A. x(n)+y(n) ↔ X(z)Y(z)
B. x(n)+y(n) ↔ X(z)+Y(z)
C. x(n)y(n) ↔ X(z)+Y(z)
D. x(n)y(n) ↔ X(z)Y(z)
Answer: B
According to the linearity property of z-transform, if X(z) and Y(z) are the z-transforms of x(n) and y(n) respectively then, the z-transform of [x(n)↔X(z)] is x(n)+y(n) is X(z)+Y(z).
17. What is the z-transform of the signal x(n) = [3(2n)-4(3n)]u(n)?
A. \(\frac{3}{1-2z^{-1}}-\frac{4}{1-3z^{-1}}\)
B. \(\frac{3}{1-2z^{-1}}-\frac{4}{1+3z^{-1}}\)
C. \(\frac{3}{1-2z}-\frac{4}{1-3z}\)
D. None of the mentioned
Answer: A
Let us divide the given x(n) into x1(n) = 3(2n)u(n) and x2(n) = 4(3n)u(n)
and x(n) = x1(n)-x2(n)
From the definition of z-transform
X1(z) = \(\frac{3}{1-2z^{-1}}\) and X2(z) = \(\frac{4}{1-3z^{-1}}\)